This report from COMPASS Research Centre at the University of Auckland examines the linkage of children to parents, grandparents and great-grandparents through birth records in the Integrated Data Infrastructure (IDI).

Citation: Milne, B. J., Li E., Sporle A. (2020). Intergenerational Analyses Using the IDI:
An update.
COMPASS Research Centre, https://www.auckland.ac.nz/content/dam/uoa/
auckland/arts/our-research/research-institutes-centres-groups/compass/
whole-population-data-analysis/Intergenerational-Links-IDI-Update.pdf
.

We have a local copy of the report available here.